Method and apparatus for measuring the birefringence autocorrelation length in optical fibers

ABSTRACT

Disclosed is a method and apparatus for determining the birefringence autocorrelation length of a fiber in a non-destructive manner. The PMD of an optical fiber is measured over a first optical spectrum. A Faraday rotation angle is measured over a second optical spectrum. The birefringence autocorrelation length is determined from the measuring of the PMD and the Faraday rotation angle.

This application is a continuation of prior application Ser. No.12/001,155 filed Dec. 10, 2007, which is incorporated herein byreference.

BACKGROUND OF THE INVENTION

The present invention relates generally to optical fibers, and moreparticularly to measuring the birefringence autocorrelation length inoptical fibers.

Fiber optic technology and fiber optic transmission systems haverevolutionized telecommunications. The main driving force behind thisrevolution is the promise of extremely high communications bandwidth. Asingle wavelength of modulated laser light can carry vast amounts ofinformation sufficient to comprise literally hundreds of thousands ofphone calls or hundreds of video channels.

In a fiber optic transmission system, information may be conveyed onmultiple optical signals, each centered on a different wavelength.Digital signals can be represented by modulating laser light, e.g., byrapidly turning the laser light on and off to represent the two possiblestates of a digital bit, i.e., “1” and “0”, or “on” and “off”. Thisprocess may be referred to as on-off-keyed modulation. Other types ofmodulation of the light also exist to convey information, including bothdigital and analog forms. The wavelength signal is generally emittedfrom a device called an optical transmitter such as a laser. In thefrequency domain, this signal is centered about a frequency of theoptical carrier, such as, for example, 193,000 GHz, but the signal mayalso include numerous frequency components spaced very closely about thenominal center frequency.

An optical signal is transmitted in a fiber optic transmission systemusing, generally, an optical transmitter, which includes a light sourceor laser, an optical fiber, optionally optical amplifier(s), and anoptical receiver. A modulated optical signal arriving at an opticalreceiver must be of sufficient quality to allow the receiver to clearlydistinguish the modulation pattern of the light sent by the transmitter(e.g., the on- and off-pulses). Noise, attenuation, and dispersion are afew of the impairments that can distort an optical signal and render theoptical) signal marginal or unusable at the receiver. The distortion ofan optical signal makes it extremely difficult or impossible for anoptical receiver to accurately detect and reconstitute the originalsignal. Some of these distortions broaden the various light pulses,potentially resulting in overlapping pulses so that it is difficult todistinguish the various pulses from each other. This may result inincreased errors in transmitting information. Conventionally, a properlydesigned optical link can maintain a sufficiently low Bit Error Rate(“BER”).

Dispersion can be a major contributor to the distortion of an opticalsignal, which may increase the BER of the optical signal. The distortioncaused by dispersion generally increases as the bandwidth of each lightpulse or data rate increases and as the optical fiber transmissiondistance increases.

One type of dispersion is Polarization Mode Dispersion (“PMD”) which isan effect related to the polarization of the optical signal. It is wellknown that monochromatic light (as from a laser) is polarized and that,for a given beam of light, the state of this polarization may beexpressed with excellent approximation in terms of two principalpolarization modes represented by two orthogonal axes that are normal tothe axis of propagation. Thus, an optical signal may be considered as asuperposition of two polarization signals, each aligned with the twoprincipal polarization modes of the fiber. As such an optical signalpropagates through the optical fiber, the two polarization signalstravel or propagate at different speeds, due to the birefringence (ordouble refraction) of that fiber. The different velocities associatedwith the two different principal polarization modes result in one of thetwo polarization signals leading the other polarization signal. Thedelay between the leading signal and the trailing signal is referred toas the Differential Group Delay (“DGD”) (denoted by τ). This speeddifference in the two polarization signals causes pulse broadening andrestricts the usable bandwidth of each optical carrier.

At each frequency, two polarization axes may be identified in a fiber—afast axis and a slow axis. Light that is polarized along the fast axishas a greater velocity than light which is polarized along the slowaxis. Light polarized along either of these axes is said to be in aprincipal polarization state. Any signal propagating in that fiber canbe expressed as a combination of signals polarized along these twoaxes—i.e. as a combination of the two principal polarization states. The“polarization state” of the light expresses how much of the light ispolarized along each of the axes as well as the relative phase betweenthese two components. The light in a pulse is usually a mixture of bothpolarizations and will therefore spread due to the velocity differencebetween the portion of the light polarized along the fast and slow axes.This velocity difference is due to the birefringence of the fiber, acharacteristic or property of the material. And just as the state ofpolarization can be thought of as having a specific orientation, so toothe local birefringence has an orientation with respect to the fiber.

In many optical fibers, not only is birefringence present, but thebirefringence is nonuniform and varies along the length of the fiber. Inother words, the local birefringence, β(z), as a function of theposition z along the fiber, may vary in either magnitude or orientation.

Many different phenomena may contribute to causing the birefringence,including, for example, asymmetrical fiber optic transmission media,mechanical stresses and strains applied to the fiber optic media, andother physical phenomena such as temperature gradients and changes. Withthese multiple phonomena affecting the birefringence in an essentiallyrandom fashion along a fiber, β becomes a statistical quantity.Depending on the number and magnitude of these sundry causes, β(z) mayvary slowly or rapidly along the fiber. Beginning at an arbitrary pointz=0 in the fiber, typically there is a small distance over which β(z)changes very little, and then there will be a distance over which it ischanging, becoming less correlated with the magnitude and orientation ofβ(0). Following that, β(z) will become completely decorrelated withβ(0). Thus, a calculation of the normalized autocorrelation of β for agiven fiber will have a specific width (e.g., when the autocorrelationfalls from 1 to ½) which corresponds to the distance along the fiber forwhich the local birefringence changes only slightly in orientation ormagnitude. This quantity is commonly referred to as the birefringenceautocorrelation length.

A fiber traditionally has to be cut in order to measure the localbirefringence β(z) or its autocorrelation length. This destructiveprocedure is necessary because the birefringence varies along thefiber's length and spatially-resolved polarization-sensitive opticaltechniques (such as polarization optical time-domain reflectometery,P-OTDR) cannot determine all types of birefringence. Specifically,P-OTDR is incapable of sensing any local circular birefringence in thefiber because the probe light must retrace its path through the exactsame local birefringence, but in the reverse direction.

In order to determine the birefringence autocorrelation length in afiber, the fiber is cut into particular sections, and the DGD, forexample, is determined for each fiber section. The fiber's DGD is thenplotted against the fiber's length to determine the relationship betweenthe fiber's DGD and its length. It has been found that when localbirefringence is constant in orientation and magnitude, the DGDassociated with the fiber is a linear function of the fiber's length.When β varies significantly over length, the relationship between afiber's DGD and its length is proportional to √{square root over (L)}.The crossover point between a linear relationship of DGD with length anda √{square root over (L)} relationship is often viewed as representingthe birefringence autocorrelation length.

The birefringence autocorrelation length is often viewed as beingextremely important in describing the polarization properties of a fiberand so there is great interest in measuring a fiber's birefringenceautocorrelation length. However, as discussed, the use of destructivetechniques to measure this value introduces a new set of problems. Forexample, connecting the many segments of a fiber after cutting it todetermine the birefringence autocorrelation length may introduceabnormalities and deformities (such as additional loss and reflections).But, even with optically perfect splices, such a procedure is very laborintensive and therefore costly.

As a result, there remains a need to determine the birefringenceautocorrelation length of a fiber in a non-destructive manner.)

BRIEF SUMMARY OF THE INVENTION

In accordance with an embodiment of the present invention, thebirefringence autocorrelation length of a fiber is determined in anon-destructive manner. The PMD of an optical fiber is measured at manyfrequencies over a first optical spectrum. A Faraday rotation angle ismeasured at multiple frequencies over a second optical spectrum for agiven magnetic field strength along the direction of the propagation ofthe light. The birefringence autocorrelation length is determined fromthe measurements of the PMD and the Faraday rotation angle. C1

In one embodiment, a beat length is determined from the measurement ofthe PMD and the Faraday rotation angle. Further, a birefringenceautocorrelation length is determined from the measuring of the PMD andthe Faraday rotation angle. In one embodiment, the birefringence islinear. A representation of the birefringence may lie on the equator ofa Poincaré sphere. In one embodiment, a derivative of the birefringenceis determined. The derivative may lie on the equator also. Thebirefringence autocorrelation length, L_(F), from the equation

${\left\langle \tau^{2} \right\rangle\left\langle \theta_{F}^{2} \right\rangle} = {\frac{16\;\beta_{F}^{2}L^{2}}{\omega_{62}}\left\lbrack {1 + \left( \frac{4\;\pi\; L_{F}}{L_{t}} \right)^{2}} \right\rbrack}$can then be determined.

These and other advantages of the invention will be apparent to those ofordinary skill in the art by reference to the following detaileddescription and the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a prior art block diagram of a material (e.g., crystal)exhibiting birefringence;

FIG. 2 is a flowchart illustrating the steps performed by a computer todetermine the polarization birefringence, the beat length, and thebirefringence autocorrelation length of a fiber in accordance with anembodiment of the present invention;

FIG. 3 is a graph of analytical results compared with computersimulations performed by using a random modulus model for thebirefringence of a fiber in accordance with an embodiment of the presentinvention;

FIG. 4 is a graph of the product τ_(rms)θ_(rms)=√{square root over (

τ²

θ_(F) ²

)} for several values of the beat length L_(B) for fiber length L=5 kmand for L_(F)=20 m in accordance with an embodiment of the presentinvention;

FIG. 5 is a graph of the product τ_(rms)θ_(rms) versus theautocorrelation length L_(F) of a fiber in accordance with an embodimentof the present invention; and

FIG. 6 is a high level block diagram of a computer system which may beused in an embodiment of the invention.

DETAILED DESCRIPTION

As described above, birefringence is a property or characteristic of afiber. Inside the fiber, there are two axes—a slow axis and a fast axis.When two pulses are sent along the fiber, each pulse may travel alongthe fiber at different speeds. As a result, the pulses have adifferential group delay (DGD) (typically measured in picoseconds) frompolarization mode dispersion (PMD). In accordance with an embodiment ofthe present invention, the PMD of a fiber is determined. The PMD of afiber may be determined using well-known techniques or devices thatmeasure the PMD of a fiber. The PMD is measured over a range of opticalfrequencies in order to determine an average PMD for the fiber. In oneembodiment, the range of frequencies at which PMD is measured is 1500 nmto 1600 nm.

Further, in accordance with an embodiment of the present invention, aFaraday rotation angle for the fiber is measured. The Faraday rotationangle (typically measured in radians) is measured by applying a magneticfield to the fiber over a range of optical frequencies. This range ofoptical frequencies may be the same as or different than the range ofoptical frequencies used to) measure the PMD of the fiber. In accordancewith an embodiment of the invention, the Faraday effect is not masked bythe birefringence of the fiber.

The autocorrelation length of the fiber is then determined from the PMDof the fiber and the Faraday rotation angle of the fiber. As a result,instead of having to cut a fiber to determine the autocorrelation lengthof the fiber, an embodiment of the present invention enables thedetermination of the autocorrelation length of the fiber without havingto destroy a fiber or a section of a fiber.

1. Fiber Birefringence, Beat Length, and Birefringence AutocorrelationLength

As described above, birefringence, or double refraction, is a complexphenomenon that occurs in materials (such as calcite and other noncubiccrystals and in some stressed plastics such as cellophane). In mostmaterials, the speed of light is the same in all directions. Thesematerials are known as isotropic. Because of their atomic structure,birefringent materials are anisotropic. Anisotropic means that the speedof light depends on its direction of propagation through the material.When a light ray is incident on such materials, the light ray may beseparated into two rays called the ordinary ray and the extraordinaryray. These rays are polarized in mutually perpendicular directions, andthey travel with different speeds. Depending on the relative orientationof the material and the incident light, the rays may also travel indifferent directions.

There is one particular direction in a birefringent material in whichboth rays propagate with the same speed. This direction is called theoptic axis of the material. The optic axis is actually a directionrather than a line in the material. Nothing unusual happens when lighttravels along the optic axis. When light is incident at an angle to theoptic axis, however, the rays travel in different directions and emergeseparated in space.

FIG. 1 is a typical block diagram of a material (e.g., crystal) 100exhibiting birefringence. A narrow beam of light 105 incident onmaterial 100 (at an angle relative to optic axis 108) is split into twobeams—ordinary ray (o ray) 110 and extraordinary ray (e ray) 115. The oray 110 and e ray 115 have perpendicular polarizations. If the material100 is rotated, the extraordinary ray 115 rotates in space.

If light is incident on a birefringent plate perpendicular to itscrystal face and perpendicular to the optic axis, the two rays travel inthe same direction but at different speeds. The number of wavelengths inthe two rays in the plate is different because the wavelengths of therays differ. The rays emerge with a phase difference that depends on thethickness of the plate and on the wavelength of the incident light.

If the material has a single optic axis, or axis of anisotropy (i.e., itis uniaxial), birefringence can be formalized by assigning two differentrefractive indices to the material for different polarizations. Thebirefringence magnitude is then defined by:Δn=n _(e) −n _(o)where n_(o) and n_(e) are the refractive indices for polarizationsperpendicular (ordinary) and parallel (extraordinary) to the optic axis.

Formally, birefringence is described by a birefringence vector {rightarrow over (β)}(z) in Stokes space, whose orientation coincides with theorientation of the slow eigenstate and whose magnitude is given by

${{\beta(z)} = {\left( {n_{1} - n_{2}} \right)\frac{\omega}{c}}},$with n₁ and n₂ being the local, frequency dependent, refractive indicescorresponding to the slow and fast eigenstates, respectively and with cdenoting the velocity of light in vacuum.

The strength of the birefringence (i.e. the magnitude of thebirefringence vector), can be equivalently represented in terms of theso-called beat length

${L_{B} = \frac{2\;\pi}{\beta}},$which is the length of a uniformly birefringent fiber needed for thepropagating signal to return to its original state of polarization. Inmost fibers, the birefringence vector, and consequently also the beatlength, change randomly along the propagation axis, thereby creating ahost of complicated phenomena. The average beat length <L_(B)> is oftenused as one of the fiber specification parameters. This averagedquantity is not an inverse of the average Birefringence

${< L_{B} > \neq \frac{2\;\pi}{< \beta >}},$but rather the average of the inverse birefringence

${< L_{B}>=\left\langle \frac{2\;\pi}{\beta} \right\rangle},$which are not the same.

The autocorrelation of the random function {right arrow over (β)}(z) hasa certain characteristic width L₀, which is (as described above) calledthe birefringence autocorrelation length. Fiber birefringence {rightarrow over (β)}(z) is typically a continuous smooth function of distancez. It can, however, be assumed that the strength of the birefringence{right arrow over (β)}(z) does not typically change much over a sectionof fiber whose length is shorter than L₀. Thus a long fiber is oftenthought of as a concatenation of a large number (N) of sections, each ofa specific finite length L_(i). The length L_(i) varies randomly aroundits average value, equal to the birefringence autocorrelation: L₀=

L_(i)

). In this representation, referred to as the “birefringence waveplatemodel”, each section i has a constant value of birefringence β_(i),which varies among the sections both in magnitude and direction. Thevalue of L₀ is typically important in describing polarization propertiesof a fiber, however, at this moment no reliable procedure exists thatcan measure the fiber birefringence autocorrelation length directly.Indirect methods utilized by fiber manufactures indicate that for modernfibers, L₀ is about tens of meters long.

The birefringence autocorrelation length L₀ is typically not the lengthused for PMD modeling purposes. Customarily, the fiber birefringenceβ(z) is expanded to the first order in frequency:{right arrow over (β)}(z)={right arrow over (β)}₀+{right arrow over(β)}₁(ω)δω,  (1)where

${{{\overset{\rightarrow}{\beta}}_{1}\left( {\omega,z} \right)} = \frac{\partial{\overset{\rightarrow}{\beta}\left( {\omega,z} \right)}}{\partial\omega}},$and the frequency independent term {right arrow over (β)}₀(z) is calledthe birefringence at the center frequency. PMD calculations are usuallymade in a reference frame of the central frequency, which rotates alongthe fiber's length following its {right arrow over (β)}₀(z) profile. Asa result, the first term {right arrow over (β)}₀(z) does not enter thePMD calculations, and each of the fiber sections has its PMD vectorequal to

${\overset{\rightarrow}{\tau}}_{i} = {{\frac{\partial}{\partial\omega}\left( {{\overset{\rightarrow}{\beta}}_{i}L_{i}} \right)} = {{\overset{\rightarrow}{\beta}}_{1i}{L_{i}.}}}$In this case, section lengths are defined based on the width L₁, whichis the autocorrelation length of the derivative of the birefringencewith frequency {right arrow over (β)}₁(z). In one embodiment, the scalesL₀ and L₁ are treated as the same.2. Magnitude of a PMD Vector and Faraday Rotation Angle for one FiberSection, the Same for Many Concatenated Sections

One section (an i^(th) section) of a fiber has a birefringence strength

$\beta_{i} = {\frac{\omega}{c}\Delta\; n}$and can be ascribed a PMD vector {right arrow over (τ)}_(i), whosemagnitude is given by:

$\begin{matrix}{{\tau_{i} = {\frac{\partial\beta_{i}}{\partial\omega} \approx {\frac{{\Delta\; n} + {\omega\frac{\partial\left( {\Delta\; n} \right)}{\partial\omega}}}{c}L_{i}} \approx {\frac{\Delta\; n}{c}L_{i}}}},} & (2)\end{matrix}$where quantity Δn is generally unknown, although it is usually assumedto depend weakly on the frequency. An external magnetic field ofstrength B, when applied along the fiber axis of the same section,causes additional circular birefringence in the section β_(i) ^(F)—aphenomenon called the Faraday Effect. One consequence of the FaradayEffect acting alone is polarization rotation of the incident polarizedlight beam by an angle θ_(i). Conventionally, the strength of theFaraday Effect is described through the magnitude of θ_(i) and is givenby:

$\begin{matrix}{{\theta_{i} = {V\frac{B\lbrack T\rbrack}{\mu_{0}}L_{i}}},} & (3)\end{matrix}$where V is an empirical constant called the Verdet constant of the fiber(V=0.75×10⁻⁶ rad/Ampere for λ=1550nm in optical fibers), B is thestrength of magnetic field measured in Tesla, and

$\mu_{0} = {4\pi \times 10^{- 7}{\frac{{Tesla} \times {meter}}{Ampere}.}}$The magnetic) field-induced birefringence is

${\beta_{i}^{F} = {2 \times V\frac{B\lbrack T\rbrack}{\mu_{0}}}},$which is equal to 6×10⁻⁵ rad/m for B=50μT (the approximate strength ofthe geomagnetic field).

For a fiber of total length L consisting of a large number (N) ofidentical concatenated sections, each of length L_(i) and birefringenceβ_(i), the frequency average of the total PMD vector magnitudeτ_(Σ)could be related to that of an individual section:

$\begin{matrix}{\tau_{\Sigma} \approx {\sqrt{N}\tau_{i}} \approx {\sqrt{\frac{L}{L_{i}}}\tau_{i}} \approx {\sqrt{{LL}_{i}}\frac{\Delta\; n}{c}}} & (4)\end{matrix}$3. Method for Measuring the Polarization Birefringence, the Beat Length,and the Birefringence Autocorrelation Length of a Fiber

FIG. 2 is a flowchart illustrating the steps performed by a computer todetermine the polarization birefringence, the beat length, and thebirefringence autocorrelation length of a fiber. The PMD of a givenoptical fiber is measured over a wide optical spectrum (i.e., formultiple wavelengths) in step 205. In step 210, the Faraday rotationangle of the fiber is measured over a wide optical spectrum (i.e., formultiple wavelengths). Some or all of the wavelengths used during themeasurement made in step 210 may be the same as or different than someor all of the wavelengths used during the measurement made in step 205.In step 215, the birefringence, beat length, and/or birefringenceautocorrelation length is then computed from the previous measurements(i.e., from the measurements of the PMD and the Faraday rotation angle.Thus, these fiber characteristics can now be measured in anon-destructive and reliable manner.

4. Theory of the Faraday Effect in a Linear Birefringence CompensatedLong Fiber Link

Let the birefringence of a long fiber be represented by the Stokesvectors {right arrow over (β)}₀(z). The birefringence caused by theFaraday effect under the influence of the geomagnetic field can berepresented by {right arrow over (β)}_(F)(z). Consider a round-tripconfiguration with a Faraday mirror at the end of a fiber span of lengthL. A Faraday mirror is a device which reverses the direction ofpropagation while rotating by 180 degrees the state of polarization inthe Stokes space. Conventional metallic or glass mirrors do not exhibitthis capability. Using a folded longitudinal coordinate, with z goingfrom 0 to 2L, in the forward propagation, the longitudinal coordinate is0≦z≦L, whereas in the backward propagation the z is reversed withL<z≦2L. Apart from an inversion of the Stokes vector of the field, andthe inversion of the z axis that makes the backward reference frameleft-handed if the forward was right-handed, the reflection by theFaraday mirror produces, in the second half, an equivalent linearbirefringence {right arrow over (β)}₀(2L−z)=−{right arrow over (β)}₀(z).The polarization evolution in the backward propagation consequentlyundoes the effect of the forward propagation. On the contrary, thenon-reciprocity of the Faraday effect gives {right arrow over(β)}_(F)(2L−z)={right arrow over (β)}_(F)(z). The evolution equation forthe forward and backward propagation is such that

$\begin{matrix}{{\frac{\mathbb{d}\hat{s}}{\mathbb{d}z} = {\left\lbrack {{{\overset{\rightarrow}{\beta}}_{0}(z)} + {{\overset{\rightarrow}{\beta}}_{F}(z)}} \right\rbrack \times \hat{s}}},} & (5)\end{matrix}$where ŝ is the state of polarization (SOP) vector. The explicitdependence on the linear birefringence in the above equation may beeliminated by establishing a reference frame that rotates with thelinear birefringence, where {tilde over (s)}(z)=R(z)ŝ(z) and {tilde over(β)}_(F)(z)=R(z){right arrow over (β)}_(F)(z), such that

$\begin{matrix}{{\frac{\mathbb{d}R}{\mathbb{d}z} + {R\left\lbrack {{{\overset{\rightarrow}{\beta}}_{0}(z)} \times} \right\rbrack}} = 0.} & (6)\end{matrix}$The evolution equation of the SOP in the rotated reference framesimplifies to

$\begin{matrix}{\frac{\mathbb{d}\overset{\sim}{s}}{\mathbb{d}z} = {{{\overset{\sim}{B}}_{F}(z)} \times {\overset{\sim}{s}.}}} & (7)\end{matrix}$The formal solution of (7), after one round-trip, is a rotation of theinput SOP, that can be described by means of the rotation vector {rightarrow over (θ)}_(F), such that ŝ(2L)=exp({right arrow over(θ)}_(F)x)ŝ(0). Given the weakness of the Faraday effect caused by theearth's magnetic field, the rotation angle θ_(F)=|{right arrow over(θ)}_(F)| is expected to be small, making the following approximationlegitimate:{right arrow over (θ)}_(F≅∫) ₀ ^(2L){tilde over (β)}_(F)(z)dz.  (8)Moreover, the equality {right arrow over (β)}₀(2L−z)=−{right arrow over(β)}₀(z), along with Eq. (6) implies that R(2L−z)=R(z), in turn implyingthat {tilde over (β)}_(F)(2L−z)={tilde over (β)}_(F)(z). Thus, Eq. (8)becomes:{right arrow over (θ)}_(F≅)2∫₀ ^(L) R(z){right arrow over(β)}_(F)(z)dz.  (9)So far, no assumption about the birefringence {right arrow over (β)}₀(z)has been made. It may be linear, or elliptical, and its dependence on zcould be arbitrary as well. In one embodiment, the birefringence isassumed to be linear, implying that the vector {right arrow over(β)}₀(z) lies on the equator of the Poincare sphere and so does itsfrequency derivative {right arrow over (β)}_(w)(z). In addition, tosimplify the analysis, a circular and constant Faraday birefringence,{right arrow over (β)}_(F)(z)={right arrow over (β)}_(F)ŝ₃ is assumed.

It is possible to show that the solution of equation (7) can beexpressed in the form of a Dyson series. Posing R(0,z)=R(z), we have:

$\begin{matrix}{{{R\left( {z_{0} + z} \right)} = {I + {\sum\limits_{m = 1}^{\infty}{\left( {- 1} \right)^{m}{\int_{D_{m}}\ {{\mathbb{d}^{m}{{\zeta\left\lbrack {{{\overset{\rightarrow}{\beta}}_{0}\left( \zeta_{1} \right)} \times} \right\rbrack}\left\lbrack {{{\overset{\rightarrow}{\beta}}_{0}\left( \zeta_{2} \right)} \times} \right\rbrack}}\mspace{14mu}{\ldots\mspace{14mu}\left\lbrack {{{\overset{\rightarrow}{\beta}}_{0}\left( \zeta_{m} \right)} \times} \right\rbrack}}}}}}},} & (10)\end{matrix}$where a z-ordered domain D_(m)={z₀≦z₁≦z,z₀≦z₂≦z₁, . . .z₀≦z_(m)≦z_(m−1)}is defined. This form of R(z₀,z) may be verified byusing the expansion in equation (6). In order to derive an expressionfor the average of R(z₀,z){right arrow over (β)}_(F), [−{right arrowover (β)}₀(z′)×][−{right arrow over (β)}₀(z″)×]{right arrow over(β)}_(F)=−[{right arrow over (β)}₀(z′)·{right arrow over(β)}₀(z″)]{right arrow over (β)}_(F) is first used iteratively. It isthen assumed that {right arrow over (β)}₀(z) is isotropicallydistributed on the equator of the Stokes space. The following istherefore obtained:

R(z ₀,z){right arrow over (β)}_(F)

={1−∫_(D) ₂ d ² z

[{right arrow over (β)}₀(z ₁)·{right arrow over (β)}₀(z ₂)]

+∫_(D) ₄ d ⁴ z

[{right arrow over (β)}₀(z ₁)·{right arrow over (β)}₀(z ₂)][{right arrowover (β)}₀(z ₃)·{right arrow over (β)}₀(z ₄)]

. . . }{right arrow over (β)}_(F),  (11)where brackets denote ensemble averaging. A further assumption that canbe made on the birefringence is that, in the analytical calculations, amodel in which the modulus of the birefringence is fixed whereas theorientation is a random process. Let the correlation function of {rightarrow over (β)}₀(z) be:

{right arrow over (β)}₀(z)·{right arrow over (β)}₀(z′)

=β₀ ² C(z−z′)  (12)with C(0)=1 and C(x)=C(−x). Within this model, the averages of productsin expansion (11) factorize into products of averages and Eq. (12) canbe simplified to:

R(z ₀ ,z){right arrow over (β)}_(F)

=R(z−z ₀){right arrow over (β)}F  (13)with

$\begin{matrix}{{{R\left( {z - z_{0}} \right)} = \left\{ \begin{matrix}{1 - {\beta_{0}^{2}{\int_{D_{2}}\ {{\mathbb{d}^{2}z}\; C\left( {z_{1} - z_{2}} \right)}}} +} \\{{{\beta_{0}^{4}{\int_{D_{4}}\ {{\mathbb{d}^{4}z}\;{C\left( {z_{1} - z_{2}} \right)}{C\left( {z_{3} - z_{4}} \right)}}}} + \ldots}\mspace{11mu}}\end{matrix}\; \right\}},} & (14)\end{matrix}$equivalent to the following integro-differential equation,

$\begin{matrix}{{\frac{\mathbb{d}{R(z)}}{\mathbb{d}z} = {{- \beta_{0}^{2}}{\int_{0}^{z}\ {{\mathbb{d}z^{\prime}}{C\left( {z - z^{\prime}} \right)}{R\left( z^{\prime} \right)}}}}},} & (15)\end{matrix}$where, without loss of generality, we used z₀=0. Laplace transformingboth sides of the equation and using the convolution theorem, thefollowing is obtained:s{tilde over (R)}(s)−R(0)=−β₀ ² {tilde over (C)}(s){tilde over(R)}(s),  (16)where tilde denotes the Laplace transform and s is the variable ofLaplace. This equation, solved for {tilde over (R)}(s) with initialcondition R(0)=1, gives

$\begin{matrix}{{{\overset{\sim}{R}(s)} = \frac{1}{s + {\beta_{0}^{2}{\overset{\sim}{C}(s)}}}},} & (17)\end{matrix}$which once inverse transformed gives the solution of (15).

The statistics of {right arrow over (θ)}_(F) can be elaborated,evaluating its average square value, α(z)=

θ_(F) ²(z)

. It can be shown that

$\begin{matrix}\begin{matrix}{{a(z)} = {8\beta_{F}^{2}{\int_{0}^{z}\ {{\mathbb{d}z^{\prime}}{\int_{0}^{z^{\prime}}\ {{\mathbb{d}z^{''}}{R\left( {z^{\prime} - z^{''}} \right)}}}}}}} \\{= {8\beta_{F}^{2}{\int_{0}^{z}\ {{\mathbb{d}{z^{\prime}\left( {z - z^{\prime}} \right)}}{{R\left( z^{\prime} \right)}.}}}}}\end{matrix} & (18)\end{matrix}$Performing the Laplace transform, the following is obtained:

$\begin{matrix}{{{\overset{\sim}{a}(s)} = \frac{8\beta_{F}^{2}}{s^{2}\left\lbrack {s + {\beta_{0}^{2}{C(s)}}} \right\rbrack}},} & (19)\end{matrix}$For large z, the asymptotic behavior is determined by the limit for s→0,that is

$\begin{matrix}{{\left. {\overset{\sim}{a}(s)} \right.\sim\frac{8\beta_{F}^{2}}{s^{2}\beta_{0}^{2}L_{0}}},{s->0},} & (20)\end{matrix}$implying that, at z=L

$\begin{matrix}{{\left. \left\langle \theta_{F}^{2} \right\rangle \right.\sim\frac{8\beta_{F}^{2}L}{\beta_{0}^{2}L_{0}}},{L->\infty},} & (21)\end{matrix}$where the distance L₀={tilde over (C)}(0)=∫₀ ^(∞)C(z)dz is an effectivecorrelation length of the fiber birefringence. Based on the CentralLimit Theorem, the result of Eq. (21) is independent on the modelassumed for the birefringence, as long as this is represented by astationary process.

Assume now an oscillating double-exponential form for the correlationfunction of the birefringence

$\begin{matrix}{{C\left( {z - z^{\prime}} \right)} = {{\cos\left\lbrack {\frac{4\pi}{L_{t}}\left( {z - z^{\prime}} \right)} \right\rbrack}{{\exp\left( {- \frac{{z - z^{\prime}}}{L_{F}}} \right)}.}}} & (22)\end{matrix}$This is the case in which a constant spin is applied to the fiber duringthe drawing process to reduce the fiber PMD. This correlation functionis consistent with a rotation of {right arrow over (β)}₀ with theconstant spatial rate 2π/L_(t) to which is superimposed a random walk ofthe vector orientation. For this choice of the birefringencecorrelation, {tilde over (R)}(s) can be evaluated analytically,

$\begin{matrix}{{{\overset{\sim}{R}(s)} = \frac{\left( {s + {1/L_{F}}} \right)^{2} + \left( {4{\pi/L_{t}}} \right)^{2}}{{s\left\lbrack {\left( {s + {1/L_{F}}} \right)^{2} + \left( {4{\pi/L_{t}}} \right)^{2} + \beta_{0}^{2}} \right\rbrack} + {\beta_{0}^{2}/L_{F}}}},} & (23)\end{matrix}$and so does, in consequence, the average square angle. The asymptoticexpression (21) of

θ_(F) ²

can be specialized to the case under scrutiny by using (22) in thealgebraic evaluation of L₀, with the result:

$\begin{matrix}{\left\langle \theta_{F}^{2} \right\rangle = {{\frac{8\beta_{F}^{2}L}{\beta_{0}^{2}L_{F}}\left\lbrack {1 + \left( \frac{4\;\pi\; L_{F}}{L_{t}} \right)^{2}} \right\rbrack}.}} & (24)\end{matrix}$An estimate of the lengths over which the asymptotic regime is reachedis L>1/(Rea/(κ)) where κ is the pole of {tilde over (R)}(s)with thesmallest real part. In the limit of L_(t)<<L_(B), L_(F), inspection ofthe denominator of {tilde over (R)}(s) gives

$\begin{matrix}{{{{\left. \frac{1}{\kappa} \right.\sim\frac{L_{F}}{\beta_{0}^{2}}}\left( \frac{4\pi}{L_{t}} \right)^{2}} = \frac{4L_{B}^{2}L_{F}}{L_{t}^{2}}},} & (25)\end{matrix}$where L_(B)=2π/β₀. For example, suppose L_(B)=19 m and L_(F)=20 m. IfL_(t)=1 m, the distance over which the asymptotic behavior is reached is1/κ˜29 km, whereas it is 1/κ˜1 km if L_(t)=5 m, and shorter for longerL_(t). Another limit is L_(t)→0 and arbitrary L. In this case, {tildeover (R)}(s)≅1/s implying that, at z=L,

θ_(F) ²

≅β_(F) ²(2L)² is obtained. This result means that in the limit of veryfast spinning rate, the random linear birefringence becomes ineffectiveand the fiber behaves like an ideal isotropic fiber of length 2L (theround-trip length) having only the circular birefringence induced by theFaraday effect.

FIG. 3 shows a graph 300 of the analytical results compared withcomputer simulations performed by using the random modulus model for thebirefringence. The net dependence of the magnitude of the rotation θ_(F)on the spin period is shown in FIG. 3, where the root mean square of therotation, θ_(rms)=√{square root over (

θ_(F) ²

)}, is plotted versus the spin period, L_(t). The solid line refers tothe analytical result of Eq. (23), while the dashed line is theasymptotic expression of Eq. (24) and the circles are the results of aMonte Carlo simulation.

Simulations where made for L_(B)=19 m, L_(F)=20 m, and for the fiberlength of L=25 km. Both theory and simulation predict, for typicalvalues of the spin period in the range of tens of meters, values ofθ_(rms) which are on the same order of magnitude as the experimentalmeasurements. As expected, as L=25 km is smaller than 1/κ for L_(t)=1 m,only the first point is outside the range of validity of the asymptoticexpression.

Thus, a method and apparatus for determining the autocorrelation lengthof the fiber birefringence is disclosed. In the asymptotic regime, theexpression for the mean square DGD,

τ²

=2β₀ ²L_(F)L/ω², where ω/2π is the optical frequency, can be used toremove the dependence of Eq. (24) on the beat length as follows:

$\begin{matrix}{{\left\langle \tau^{2} \right\rangle\left\langle \theta_{F}^{2} \right\rangle} = {{\frac{16\;\beta_{F}^{2}L^{2}}{\omega^{2}}\left\lbrack {1 + \left( \frac{4\pi\; L_{F}}{L_{t}} \right)^{2}} \right\rbrack}.}} & (26)\end{matrix}$This way, for given fiber length and spin period, and for measuredvalues of PMD and rotation angle, Eq. (26) can be solved for theautocorrelation length of the fiber birefringence, L_(F).

Specifically, Eq. (26) can be used to obtain the following formula:

$\begin{matrix}{\frac{L_{F}}{L_{t}} = {{\frac{1}{4\pi}\left\lbrack {\frac{\omega^{2}\left\langle \tau^{2} \right\rangle\left\langle \theta_{F}^{2} \right\rangle}{16\beta_{F}^{2}L^{2}} - 1} \right\rbrack}^{1/2}.}} & (27)\end{matrix}$

FIG. 4 shows a graph 400 of the product τ_(rms)θ_(rms)=√{square rootover (

τ²

θ_(F) ²

)} for several values of the beat length L_(B) for fiber length L=5 kmand for L_(F)=20 m. Specifically, in FIG. 4, the productτ_(rms)θ_(rms)=√{square root over (

τ²

θ_(F) ²

)} is plotted versus the spin period by the solid lines for the valuesof the beat length, while the asymptotic expression of Eq. (26) isplotted by a dashed line. Dots refer to a Monte Carlo simulation madefor L_(B)=15 m. The plot confirms the independence of the productτ_(rms)θ_(rms) on the beat length, already for small values of the spinperiod.

Periodically Spun Fibers

PMD of fibers can be further reduced by optimizing the spin pattern,which can be done by making the angular velocity of the spinner timedependent. The rotation of the spinner can be described by the periodicfunction A(z), that is assumed here to be periodic. The period Z givesthe length of the preform drawn by the spinner to repeat its angularposition and velocity. Recall the asymptotic expression for the Faradayrotation angle, given by Eq. (21) above:

$\begin{matrix}{{\left\langle \theta_{F}^{2} \right\rangle \approx \frac{8\beta_{F}^{2}L}{\beta_{0}^{2}{\overset{\sim}{C}}_{0}}},} & (28)\end{matrix}$with

${\overset{\sim}{C}\;}_{0} = {{Z^{- 1}\left\lbrack {1 - {\exp\left( {- \frac{Z}{L_{F\;}}} \right)}} \right\rbrack}^{- 1} \times {\int_{0}^{z}{\int_{0}^{z}{{\mathbb{e}}^{- \frac{u}{L_{F}}}\cos\left\{ {2\begin{bmatrix}{{A(z)} -} \\{A\left( {z - u} \right)}\end{bmatrix}} \right\}{\mathbb{d}u}{{\mathbb{d}z}.}}}}}$The asymptotic expression for the mean square DGD can be written asfollows:

$\begin{matrix}{< \tau^{2}>={\frac{2\beta_{0}^{2}L_{F}L}{\omega^{2}}\frac{1}{L_{F}}\frac{{\overset{\sim}{C}}_{0}^{2} + {\overset{\sim}{S}}_{0}^{2}}{{\overset{\sim}{C}}_{0}}}} & (29)\end{matrix}$with

${\overset{\sim}{S}}_{0} = {{Z^{- 1}\left\lbrack {1 - {\exp\left( {- \frac{Z}{L_{F}}} \right)}} \right\rbrack}^{- 1} \times {\int_{0}^{z}{\int_{0}^{z}{{\mathbb{e}}^{- \frac{u}{L_{F}}}\sin\left\{ {2\begin{bmatrix}{{A(z)} -} \\{A\left( {z - u} \right)}\end{bmatrix}} \right\}{\mathbb{d}u}{{\mathbb{d}z}.}}}}}$As described above, multiplying the two equations, an expressionindependent of the beat length is obtained:

$\begin{matrix}{{\left( \theta_{F}^{2} \right)\left( \tau^{2} \right)} = {\frac{16\beta_{F}^{2}L^{2}}{\omega^{2}}\left( {1 + \frac{{\overset{\sim}{S}}_{0}^{2}}{{\overset{\sim}{C}}_{0}^{2}}} \right)}} & (30)\end{matrix}$Inversion of Eq. (30) in favor of L_(F) is not straightforward. Ingeneral, for a given spin pattern, Eq. (30) can be plotted versus theautocorrelation length of the fiber birefringence. Thus, the value ofL_(F) for which the plot meets the experimentally measured value of<θ_(F) ²><τ²> represents the wanted estimate of the correlation length.

Below is a numerical example. Consider the following spin function:

$\begin{matrix}{{A(z)} = {A_{0}\left\lbrack {{\sin\left( {\frac{2\pi}{Z}z} \right)} + {2\;{\sin\left( {\frac{4\pi}{Z}z} \right)}}} \right\rbrack}} & (31)\end{matrix}$The spin amplitude can be A₀=1.2 rad and the spin period Z=4 m. Forthese parameter values, Eq. (30) is plotted versus L_(F) in FIG. 5.Specifically, FIG. 5 shows a graph 500 of the product τ_(rms)θ_(rms))versus the autocorrelation length L_(F). Corresponding to the exemplaryvalues (L, τ_(rms), θ_(rms))≅(25 km; 0.2 ps; 0.2 rad), the productτ_(rms)θ_(rms)≅0.04 is evaluated, which gives the two estimates L_(F)≅6m or L_(F)≅13 m.

It should be pointed out that, for some classes of spin functions, theintegral in the definition of {tilde over (S)}₀ vanishes, thus theasymptotic value of (θ_(F) ²)(τ²) becomes independent of thebirefringence autocorrelation length. Nonetheless, this property, whileinvalidating the use of angle measurements for estimating L_(F), allowsthe measuring of the birefringence induced by the geomagnetic fieldβ_(F). For {tilde over (S)}₀0 this can be done by solving Eq. (30) infavor of β_(F), which gives β_(F)=ωθ_(rms)τ_(rms)/4L. B2

FIG. 6 shows a high level block diagram of a computer that may be usedto perform the calculations described above. Computer 600 contains aprocessor 604 which controls the overall operation of the computer byexecuting computer program instructions which define such operation. Thecomputer program instructions may be stored in a storage device 612(e.g., magnetic disk, database) and loaded into memory 610 whenexecution of the computer program instructions is desired. Thus, thecomputer operation will be defined by computer program instructionsstored in memory 610 and/or storage 612 and the computer will becontrolled by processor 604 executing the computer program instructions.Computer 600 also includes one or more input network interfaces 606 forcommunicating with other devices via a network (e.g., the Internet).Computer 600 also includes one or more output network interfaces 606 forcommunicating with other devices. Computer 600 also includesinput/output 608 which represents devices which allow for userinteraction with the computer 600 (e.g., display, keyboard, mouse,speakers, buttons, etc.). One skilled in the art will recognize that animplementation of an actual computer will contain other components aswell, and that FIG. 6 is a high level representation of some of thecomponents of such a computer for illustrative purposes.

The foregoing Detailed Description is to be understood as being in everyrespect illustrative and exemplary, but not restrictive, and the scopeof the invention disclosed herein is not to be determined from theDetailed Description, but rather from the claims as interpretedaccording to the full breadth permitted by the patent laws. It is to beunderstood that the embodiments shown and described herein are onlyillustrative of the principles of the present invention and that variousmodifications may be implemented by those skilled in the art withoutdeparting from the scope and spirit of the invention. Those skilled inthe art could implement various other feature combinations withoutdeparting from the scope and spirit of the invention.

1. A method of determining characteristics of a fiber comprising:measuring, at a computer, polarization mode dispersion (PMD) of anoptical fiber; measuring a rotation angle; and determining birefringenceautocorrelation length from the measuring of the PMD and the rotationangle, wherein the birefringence autocorrelation length, L_(F), isdetermined by the following equation:${{\left\langle \tau^{2} \right\rangle\left\langle \theta_{F}^{2} \right\rangle} = {\frac{16\beta_{F}^{2}L^{2}}{\omega^{2}}\left\lbrack {1 + \left( \frac{4\pi\; L_{F}}{L_{t}} \right)^{2}} \right\rbrack}},$ where τ represents a differential group delay, θ_(F) represents arotation angle, β_(F) represents a fiber birefringence, L represents afiber length, and ω represents a frequency and L_(t) represents a spinperiod.
 2. The method of claim 1 wherein the birefringenceautocorrelation length is determined in a non-destructive manner.
 3. Themethod of claim 1 further comprising determining a beat length.
 4. Themethod of claim 1 further comprising determining a polarizationbirefringence.
 5. An article of manufacture including a non-transitorycomputer-readable medium having instructions stored thereon that, ifexecuted by a computing device, cause the computing device to performoperations comprising: measuring polarization mode dispersion (PMD) ofan optical fiber; measuring a rotation angle; and determiningbirefringence autocorrelation length from the measuring of the PMD andthe rotation angle, wherein the birefringence autocorrelation length,L_(F), is determined by the following equation:${{\left\langle \tau^{2} \right\rangle\left\langle \theta_{F}^{2} \right\rangle} = {\frac{16\beta_{F}^{2}L^{2}}{\omega^{2}}\left\lbrack {1 + \left( \frac{4\pi\; L_{F}}{L_{t}} \right)^{2}} \right\rbrack}},$ where τ represents a differential group delay, θ_(F) represents arotation angle, β_(F) represents a fiber birefringence, L represents afiber length, and ω represents a frequency and L_(t) represents a spinperiod.
 6. The article of manufacture of claim 5 wherein thebirefringence autocorrelation length is determined in a non-destructivemanner.
 7. The article of manufacture of claim 5 wherein the operationsfurther comprise determining a beat length.
 8. The article ofmanufacture of claim 5 wherein the operations further comprisedetermining a polarization birefringence.